Show that cauchy sequences are not topological terms, find a counterexample?

Question

I need to show that the expression 'is a cauchy sequence' is not a topological expression.

I could show this by finding a homeomorphic function $\phi$ such that a Cauchy sequence is mapped onto a sequence which is not Cauchy.

I have no idea how to find this example. Could someone give me some pointers?

Answer 1

Let $X = (0,\infty)$, and consider: $\Phi: X \longrightarrow X$, $\Phi(t) = 1/t$.

Then, $\Phi$ is a homeomorphism. However, $\{1/n\}_n$ is cauchy in X and $\{f(x_n) = n \}_n$ is not Cauchy.